Forcing a classification of non-torsion Abelian groups of size at most 2 c with non-trivial convergent sequences
Matheus Koveroff Bellini, Vinicius de Oliveira Rodrigues, Artur Hideyuki Tomita
aa r X i v : . [ m a t h . GN ] J un FORCING A CLASSIFICATION OF NON-TORSION ABELIAN GROUPSOF SIZE AT MOST c WITH NON-TRIVIAL CONVERGENTSEQUENCES
MATHEUS KOVEROFF BELLINI, VINICIUS DE OLIVEIRA RODRIGUES,AND ARTUR HIDEYUKI TOMITA
Abstract.
We force a classification of all the Abelian groups of cardinality at most 2 c thatadmit a countably compact group with a non-trivial convergent sequence. In particular, weanswer (consistently) Question 24 of Dikranjan and Shakhmatov [10] for cardinality at most2 c , by showing that if a non-torsion Abelian group of size at most 2 c admits a countablycompact Hausdorff group topology, then it admits a countably compact Hausdorff grouptopology with non-trivial convergent sequences. Introduction
Some history.
There are three natural questions concerning countably compactAbelian group without non-trivial convergent sequences that can be asked separately orjointly.1) What groups admit such topologies?2) How large they can be?3) Do they exist in ZFC?Question 1 was solved under Martin’s Axiom in [11] for Abelian groups of cardinality c .This was later improved in [2] under the use of c selective ultrafilters.Dikranjan and Shakhmatov [9] used a forcing model to classify all Abelian groups ofcardinality at most 2 c that admit a countably compact group topology (without non-trivialconvergent sequences).Question 2 was solved for torsion groups in Castro-Pereira and Tomita [8]. They classified,using some cardinal arithmetic and the existence of a selective ultrafilter p , all the torsiongroups that admit a p -compact group topology (without non-trivial convergent sequences).This gave the first arbitrarily large countably compact groups without non-trivial convergentsequences. Earlier examples had their size limited to 2 c Question 3 is the best known question in the subject. It has been finally answered by M.Hrusak, U. A. Ramos-Garcia, J. van Mill and S. Shelah in [19]. The main new ingredient inthe ZFC construction is the use of a clever filter which takes care of the combinatorics thatguarantee the existence of accumulation points. This new idea has yet two limitations, theconstruction depends on the use of a group of finite order and the example has cardinality c . It is not yet known if the example could be improved to obtain an example of cardinalitystrictly greater than c . These questions appear in their article: Mathematics Subject Classification.
Primary 54H11, 22A05; Secondary 54A35, 54G20.The first listed author has received financial support from FAPESP 2017/15709-6.The second listed author has received financial support from FAPESP 2017/15502-2.The third listed author has received financial support from FAPESP 2016/26216-8.Keywords: countable compactness, convergent sequences, topological group.
Problem 1.1.
Are there larger countably compact groups without non-trivial convergentsequences in ZFC?
Problem 1.2.
Is there a torsion-free countably compact group without non-trivialconvergent sequences in ZFC?Due to this last question, we are still far from a classification (in ZFC) of countablycompact Abelian groups with no convergent sequences of cardinality c . And due to the firstquestion, we are even farther from a classification of group of cardinality ≤ c in ZFC.To these questions, one can add an interesting question from Dikranjan and Shakhmatov[10], Question 24 whether an infinite group admitting a countably compact Hausdorff grouptopology can be endowed with a countably compact Hausdorff group topology that contains anon-trivial convergent sequence. It is well-known the difficult to construct countably compactgroups without nontrivial convergent sequences, but once there was progress to classify them,it became natural to ask how hard would it be to add a convergent sequence. They alsoasked a similar question for pseudocompact groups, which was answered by Galindo, Garcia-Ferreira and Tomita [12].In [12], it was also noted that, in ZFC, if a torsion Abelian group admits a countablycompact group topology then it admits a countably compact group topology with an infinitecompact metric subgroup, so, in particular, with a non-trivial convergent sequence. Thus,the real difficult for Dikranjan and Shakhmatov’s question is about non-torsion groups.Using the example in [8], it is consistent that every torsion group admits a countablycompact group topology if and only if admits a countably compact group topology withoutnon-trivial convergent sequences if and only if it admits a countably compact group topologywith a non-trivial convergent sequence.Boero, Garcia-Ferreira and Tomita [6] showed that the existence of c selective ultrafiltersimplies that the free Abelian group of size c admits a group topology that makes it countablycompact with a non-trivial convergent sequence. Bellini, Boero, Castro-Pereira, Rodriguesand Tomita [1] showed from p = c that every non-torsion Abelian group of cardinality c that admits a countably compact group topology also admits one which in addition has aconvergent sequence. Bellini, Boero, Rodrigues and Tomita [2] showed from the existenceof 2 c selective ultrafilters that every Abelian group of cardinality c as above admits 2 c non-homeormorphic such topologies with or without convergent sequences and any given weightof cardinality between c and 2 c . It also provided a countably compact group topology, onewith and other without non-trivial convergent sequences, for some (but not all) non-torsiongroups of cardinality 2 c .In their forcing model, Dikranjan and Shakhmatov [9] showed a non-torsion Abelian groupof cardinality at most 2 c admits a countably compact group topology (without non-trivialconvergent sequences) if and only if the free rank of G is ≥ c and, for all d, n ∈ N with d | n ,the group dG [ n ] is either finite or has cardinality ≥ c . Forcing is used to prove the ‘only if’of the equivalence, whereas the other implication holds in ZFC.We will use here the ZFC implication. In the forcing examples to produce groups withoutnon-trivial convergent sequences it is used some ideas closely related to the construction ofHFD groups. We have to proceed differently to make a sequence converge.1.2. Basic results, notation and terminology.
We recall that a T topological space is countably compact if every infinite subspace has an accumulation point in the space. ORCING COUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 3
The following definition was introduced in [4] and is closely related to countablecompactness.
Definition 1.3.
Let p be a free ultrafilter on ω and let s : ω → X be a sequence in atopological space X . We say that x ∈ X is a p -limit point of s if, for every neighborhood U of s , { n ∈ ω : s ( n ) ∈ U } ∈ p .We say a topological space is p -compact if every sequence into it has a p -limit point. (cid:3) If X is a Hausdorff space a sequence s has at most one p -limit point x and we write x = p -lim s .The set of all free ultrafilters on ω will be denoted by ω ∗ . It is not difficult to show that a T topological space X is countably compact if and only if, each sequence in X there exists p ∈ ω ∗ such that s has a p -limit point. There are several similarities between the theories oflimits of sequences and of p -limits. Proposition 1.4. If p ∈ ω ∗ and ( X i : i ∈ I ) is a family of topological spaces, then( y i ) i ∈ I ∈ Q i ∈ I X i is a p -limit point of a sequence (( x ni ) i ∈ I : n ∈ ω ) in Q i ∈ I X i if, andonly if, y i = p − lim( x ni : n ∈ ω ) for every i ∈ I . (cid:3) The following proposition is straightforward to prove. A proof can be found in [14](Theorem 3.54).
Proposition 1.5.
Let X , Y be topological spaces and f : X → Y be a continuous function, s : ω → X be a sequence in X and p ∈ ω ∗ . It follows that if x = p -lim( s n : n ∈ ω ), then f ( x ) = p -lim( f ( s n ) : n ∈ ω ). (cid:3) Since + and − are continuous functions in topological groups, it follows from the twoprevious propositions that: Proposition 1.6.
Let G be a topological group and p ∈ ω ∗ .(1) If ( x n : n ∈ ω ) and ( y n : n ∈ ω ) are sequences in G and x, y ∈ G are such that x = p − lim( x n : n ∈ ω ) and y = p − lim( y n : n ∈ ω ), then x + y = p − lim( x n + y n : n ∈ ω );(2) If ( x n : n ∈ ω ) is a sequence in G and x ∈ G is such that x = p − lim( x n : n ∈ ω ),then − x = p − lim( − x n : n ∈ ω ). (cid:3) A pseudointersection of a family G of sets is an infinite set that is almost contained inevery member of G . We say that a family G of infinite sets has the strong finite intersectionproperty (SFIP, for short) if every finite subfamily of G has infinite intersection.We denote the set of positive natural numbers by N , the integers by Z , the rationals by Q and the reals by R . The unit circle group T will be identified with the metric group ( R / Z , δ )where δ is given by δ ( x + Z , y + Z ) = min {| x − y + a | : a ∈ Z } for every x, y ∈ R . Given asubset A of T , we will denote by δ ( A ) the diameter of A with respect to the metric δ . Theset of all non-empty open arcs of T will be denoted by B .Let X be a set and ( G, + ,
0) be a group. We denote by G X the product Q x ∈ X G x where G x = G for every x ∈ X . The support of g ∈ G X is the set { x ∈ X : g ( x ) = 0 } , whichwill be designated as supp g . The set { g ∈ G X : | supp g | < ω } will be denoted by G ( X ) . If f : ω → G ( X ) is a sequence, then supp f = S n ∈ ω supp f ( n ).The torsion part T ( G ) of an Abelian group G is the set { x ∈ G : nx = 0 for some n ∈ N } .Clearly, T ( G ) is a subgroup of G . For every n ∈ N , we put G [ n ] = { x ∈ G : nx = 0 } . Inthe case G = G [ n ], we say that G is of exponent n provided that n is the minimal positiveinteger with this property. The order of an element x ∈ G will be denoted by o( x ). M. K. BELLINI, V. O. RODRIGUES, AND A. H. TOMITA
A non-empty subset S of an Abelian group G is said to be independent if 0 S and, givendistinct elements s , . . . , s n of S and integers m , . . . , m n , the relation m s + . . . + m n s n = 0implies that m i s i = 0, for all i . The free rank r ( G ) of G is the cardinality of a maximalindependent subset of G such that all of its elements have infinite order. It is easy to verifythat r ( G ) = | G/T ( G ) | if r ( G ) is infinite.An Abelian group G is called divisible if, for each g ∈ G and each n ∈ N \ { } , there exists x ∈ G such that nx = g . If n ∈ N , we denote by G [ n ] the set { x ∈ G : nx = 0 } .The proof of the next three results are well known basic results of the theory of divisiblegroups can be found in [18]. Proposition 1.7.
Let G be an Abelian group, H be a subgroup of G , ˜ G be a divisible groupand f : H → ˜ G be a group homomorphism. There exists a group homomorphism F : G → ˜ G such that F ↾ H = f . (cid:3) The group Q / Z is called the quasicyclic group . Theorem 1.8.
An Abelian group is divisible if and only if, it is isomorphic to a direct sumof copies of Q and of quasicyclic groups. (cid:3) Theorem 1.9.
Every Abelian group is isomorphic to a subgroup of a divisible group.2.
The groups for the immersion
We present some of the notation that will be used throughout this article.We fix a partition { P , P } of c such that | P | = | P | = c , and ω + 1 ⊆ P . Let { R , R } be a partition of 2 c \ c such that | R | = | R | = 2 c . Define U = Q ( R ) ⊕ Q ( R ) and U = ( Q / Z ) ( R ) ⊕ Q ( R ) .We define W = ( Q / Z ) ( P × ω ) ⊕ Q ( P ) ⊕ U . We also let X = Q ( P × ω ) ⊕ Q ( P ) ⊕ U .Throughout this work, the main group will be X = ⊕ n> ,m ≥ ( Q / Z ) ( C n,m × m ) ⊕ Q ( P ) ⊕ U ,where { C n,m : n > , m ≥ } is a partition of P into pieces of cardinality c and C = S m,n> C n,m is such that P \ C = S n> C n, has cardinality c . We will also define C ⊆ P with | C | = c , ω ⊆ C and | P \ C | = c . To simplify the notation of the forcinglater we will also partition C as { C ,m : m > } and let C , = P \ C .2.1. Structure of the article.
We use forcing to construct an injective grouphomomorphism Φ : X → T c . The range of this homomorphism will be countably compactand have convergent sequences. Each forcing condition will be a partial countable piece ofthis homomorphism, and the existence of suitable conditions was proved in [1].Of course, not every subgroup of X is countably compact. However, we show that if H is a group such that 2 c ≥ | H | = r ( H ) = c and for all d, n ∈ N with d | n , the group dH [ n ]is either finite or has cardinality at least c , then it is isomorphic to a subgroup of X thatis countably compact and has convergent sequences considering the subspace topology of X generated by Φ. To show that such a copy exists, we define the concept of large subgroup of X , which was inspired by the concept of nice immersion we defined in [1]. This concludesthe classification.The convergent sequences will be the sequences ( n ! S χ n : n ∈ ω ) in G ⊆ X (identifying G with its copy in X ), for each positive integer S and they will converge to 0. ORCING COUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 5
More notation.
Given w ∈ W or w ∈ X , x ∈ ( P × ω ) ∪ R and y ∈ P ∪ R , wedenote by w ( x ) the x -th coordinate of w and w ( y ) the y -th of w , so the functions w → w ( x )and w → w ( y ) are the natural projections.We also fix well defined numerators and denominators for fractions: if r ∈ Q / Z , then p ( r )and q ( r ) are the unique integers p, q such that q >
0, gcd( p, q ) = 1, 0 ≤ p < q and r = pq + Z .Likewise, if r ∈ Q , p ( r ) and q ( r ) are the unique integers p, q such that q >
0, gcd( p, q ) = 1and r = pq .Given w ∈ W (or w ∈ X ), we denote by w and w the unique elements of W (or X ) suchthat supp w ⊆ (( P × ω ) ∪ R ), supp w ⊆ P ∪ R and w = w + w , that is, w → w isthe natural projections into ( Q / Z ) (( P × ω ) ∪ R ) (or Q (( P × ω ) ∪ R ) ) and w → w is the naturalprojection into Q ( P ∪ R ) . Also, we call w , and w , the natural projections of w onto Q ( ω ) and Q (( P ∪ R ) \ ω ) , respectively.We also define p ( w ) = max {| p ( w ( z )) | : z ∈ supp w } and q ( w ) = max { q ( w ( z )) : z ∈ supp w } if w = 0. We define p (0) = 0 and q (0) = 0.Similarly, given g : ω → W (or X ), we define g , g , g , and g , . So g = g + g = g + g , + g , , supp g ⊆ (( P × ω ) ∪ R ), supp g ⊆ P ∪ R , supp g , ⊆ ω andsupp g , ⊆ ( P ∪ R ) \ ω ; where supp h = S { supp h ( k ) : k ∈ ω } for a sequence h .It will be useful to be able to easily transform an element of X into an element of W . Thus,given w ∈ X , we define [ w ] as the unique element of W such that for every x ∈ ( P × ω ) ∪ ( R ),[ w ]( x ) = w ( x ) + Z and for every y ∈ P ∪ R , [ w ]( y ) = w ( y ). Clearly, the function w → [ w ]is a group homomorphism from X onto W . Given a function g : ω → X , we also define[ g ] : ω → W be given by [ g ]( n ) = [ g ( n )] for every n ∈ ω .3. Types of sequences
Associating sequences to a type.
We will make some minor adjustments to thedefinition of the types defined in [1] to be able to work with larger groups. Basically, wedefine all the types on W as in [1]. However, the group W used in [1] is not the same group W we are using in this article, since there W does not have the component U .In this section we define the 11 types of sequences related to a a subgroup G of W andstate the theorem that every sequence is related to one of them.The name of the sequences which are of one of these 11 types of sequences for G (whichwill be defined in the following subsections) by H G .The main result we will state in this section is the following, which, in particular, impliesthat when working with the existence of accumulation points for a sequence, by passing toa subsequence it is enough to guarantee the existence of an accumulation point for the 11types and the convergence of the sequence of ( n ! S χ n : n ∈ ω ) to 0 for each positive integer S .The proof is the same as the proof of Theorem 3.1. of [1] and is omitted, although the W isdifferent. Theorem 3.1.
Let f : ω → W is given by f ( n ) = n ! χ n for every n ∈ ω . Let G be asubgroup of W containing (0 , χ n ) for every n ∈ ω . Let g : ω → X with [ g ] ∈ G ω .Then there exists h : ω → X such that h ∈ H G or [ h ] is constant and in G , c ∈ X with[ c ] ∈ G , F ∈ [ ω ] <ω , p i , q i ∈ Z with q i = 0 for every i ∈ F , ( j i : i ∈ F ) increasing enumerationsof subsets of ω and j : ω → ω strictly increasing such that g ◦ j = h + c + X i ∈ F p i q i f ◦ j i M. K. BELLINI, V. O. RODRIGUES, AND A. H. TOMITA with q i ≤ j i ( n ) for each n ∈ ω and i ∈ F (which implies [ q i f ◦ j i ] ∈ G ω since q i | (( j i ( n ))!)for each i ∈ F and n ∈ ω ). (cid:3) The types.
Each type uses some point of the support and the supports are divided inthree groups.
Definition 3.2 (The types related to ( R ∪ P ) \ ω ) . Let G be a subgroup of W . We definethe first three types of sequence (with respect to G ) as follows: Let g : ω → X be such that[ g ( n )] ∈ G , for every n ∈ ω .We say that g is of type 1 if supp g , ( n ) \ ∪ m
We say that g is of type 11 if the family { [ g ( n )] : n ∈ ω } is an independent family whoseelements have a fixed order k , for some positive integer k . (cid:3) Notice that if G is a subgroup of X then H G = G ω ∩ H X . We also set H = H X .The following lemma is easy to verify and left to the reader. Lemma 3.5.
Being a sequence of one of the types is absolute for transitive models of ZFC.4.
The Partial order and the topology in the forcing extension
In this section we define the forcing poset and prove its basic properties.4.1.
Countable homomorphisms.
In this subsection, we will state a theorem thatguarantees (in ZFC) the existence of partial homomorphisms defined on countable subgroupsof X . The statement of this theorem is very similar to Proposition 4.3. of [1] and has acompletely analogous proof, therefore we omit it. The only difference is that in [1] thecountable subset of H was not indexed and that proposition was originally stated for 1-1indexations, but the proof is the same if one allows any sequence. Proposition 4.1.
Let E be a countable subset of 2 c containing ω , e ∈ X E with e = 0, acountable { g k : k ∈ ω } ⊆ H X E and A k infinite subsets of ω for each k ∈ ω .Fix a family ( c k : k ∈ ω ) of elements of X E such that [ c k ] ∈ X E , c k is a non torsion elementif g k is of one of types from 1 to 10, and [ c k ] has the same order as [ g k ] if g k is of type 11.Then there exists a homomorphism ρ : X E → T such that:(1) ρ ( e ) = 0,(2) for each k ∈ ω , there exists B k ⊆ A k infinite such that ( ρ ([ g k ( n )])) n ∈ B k converges to ρ ([ c k ]), and(3) (cid:0) ρ (cid:0) n ! S χ n (cid:1) : n ∈ ω (cid:1) converges to 0 ∈ T , for every integer S >
Definition 4.2.
We define P as the set of the tuples of the form ( E, α, φ, G , c, A ) such that: • E is a countable subset of 2 c containing ω , • α < c , • G = ( G n,m : { ( n, m ) : n ≥ , m > } ) is such that each G n,m is a countable subset of H , where the types are defined with respect to X E . If n = 1, the elements of G n,m are sequences of types 1-10. If not, they are all of type 11 and order n . • A = ( A n,m,g : n ≥ , m > , g ∈ G n,m ) is such that each A n,m,g is an infinite subset of ω , • c = ( c n,m,g : n ≥ , m > , g ∈ G n,m ) is a family of elements of X E , • if n, m ≥ g ∈ G n,m , c n,m,g is an element of order n with c n,m,g = [ n χ ( µ, ] forsome µ ∈ C n,m , • if m ≥ g ∈ G ,m , c ,m,g = [ χ ( µ ) ] for some µ ∈ C ,m , • φ : X E → T α is an homomorphism, • ( φ ([ g ( k )])) k ∈ A n,m,g converges to φ ( c n,m,g ) for each n ≥ , m > • ( φ ([ N χ n )]) n ∈ ω converges to 0 ∈ X , for every natural N ≥ E, α, φ, G , c, A ) ≤ ( E ′ , α ′ , φ ′ , G ′ , c ′ , A ′ ) if:(1) E ⊇ E ′ (2) α ≥ α ′ M. K. BELLINI, V. O. RODRIGUES, AND A. H. TOMITA (3) G n,m ⊇ G ′ n,m for every n ≥ m > c n,m,g = c ′ n,m,g for each n ≥ m > g ∈ G ′ n,m (5) A n,m,g ⊆ ∗ A ′ n,m,g for each n ≥ m > g ∈ G ′ n,m (6) For every ξ < α ′ and a ∈ X E ′ , φ ( a )( ξ ) = φ ′ ( a )( ξ ).Given p ∈ P , we may denote its components by E p , α p , φ p , G p , c p and A p .If G is a generic filter over P then the generic homomorphism defined by G is the mappingΦ of domain S { dom φ p : p ∈ G } into T c defined by φ ( · )( ξ ) = S { φ p ( · )( α ) : p ∈ G } . In otherwords, if p ∈ G , a ∈ X E p and ξ < α p , then Φ( a )( ξ ) = φ p ( a )( ξ ). (cid:3) Of course, we must see that the generic homomorphisms are really well definedhomomorphisms into T c . We will see later that by assuming CH in the ground model, P is ω closed and has the ω -c.c., therefore it preserves cardinals and c . We reserve the restof this section to prove this fact. Proposition 4.3.
Let e ∈ X be a non-zero element. Then C e = { p ∈ P : e ∈ X E p , φ p ( e ) =0 } is open and dense in P . Proof.
Openness is clear. Fix p ∈ P . We will define an extension q ≤ p that is an elementof C e .Let E q = E p ∪ supp e and α q = α p + 1. Extend φ p : X E p → T α p to a homomorphism φ : X E q → T α p using divisibility. Apply Proposition 4.1 with { ( n, m, g ) : g ∈ G p,n,m } , { A pn,m,g : g ∈ G pn,m } and { c pn,m,g : g ∈ G pn,m } . Then there exists ρ : X E q → T such that(1) ρ ( e ) = 0,(2) for each ( n, m, g ) with n ≥ m > g ∈ G pn,m , there exists B n,m,g ⊆ A pn,m,g infinite such ( ρ ([ g ( k )])) k ∈ B n,m,g converges to ρ ([ c pn,m,g ]) and(3) ( ρ ([ n ! S χ n ]) : n ∈ ω ) converges to 0 ∈ T , for every positive integer S .Set G qn,m = G pn,m for each n ≥ m >
1. Set c qn,m,g = c pn,m,g , A qn,m,g = B n,m,g for each g ∈ G pn,m with n ≥ m > φ q = φ p⌢ ρ .Then q ≤ p and q ∈ C e . (cid:3) Proposition 4.4.
For each α < c the set A α = { p ∈ P : α p > α } is an open dense subsetof P . Proof.
Fix p ∈ P . If α < α p then p ∈ A α . So suppose that α ≥ α p .We will define q . Set E q = E p , α q = α + 1, G qn,m = G pn,m , c qn,m,g = c pn,m,g , A qn,m,g = A pn,m,g .Let ρ : X E p −→ { } [ α p ,α q [ and φ q = φ p⌢ ρ .Then q ≤ p and q ∈ A α . (cid:3) Proposition 4.5.
The partial order P is ω -closed. Proof.
Fix a decreasing sequence ( p t : t < ω ). Write p t = ( E t , α t , φ t , G t , c ξ , A t ). We define acommon extension r as follows:Let E r = S { E t : t < ω } , G rn,m = S {G tn,m : t ∈ ω } for each n ≥ m >
1. For each n ≥ m > g ∈ G rn,m , define c rn,m,g = c tn,m,g for some (every) t such that g ∈ G tn,m (thevalue does not depend of t ). Fix A rn,m,g a pseudointersection of { A tn,m,g : g ∈ G p t ,n,m } .Let α r = sup { α t : t < ω } Given ξ < α and a ∈ X E r = S t<ω X E t , let φ ( a )( ξ ) = φ t ( a )( ξ ) for some (every) t such that a ∈ X E t and α t > ξ . (cid:3) Proposition 4.6.
The partial order P has the c + -cc. ORCING COUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 9
Proof.
Fix an arbitrary subset Q of P of cardinality c + . We show that there Q has a subsetof c + -many pairwise compatible elements.Fix Q ⊆ Q of cardinality c + and α < c such that α p = α q for every p, q ∈ Q .Using the ∆-system Lemma, there exists Q ⊆ Q of cardinality c + such that { E p : p ∈Q } is a ∆-system of root ˜ E . Furthermore, using the fact that ˜ E ω has cardinality at most c , it follows that there exists Q ⊆ Q of cardinality c + such that φ p | X ˜ E = φ q | X ˜ E for every p, q ∈ Q .For each p ∈ Q , let J p = { ( n, m, g ) : n, m ≥ , g ∈ G pn,m } for each p ∈ Q . Using the∆-system Lemma, we can find Q ⊆ Q of cardinality c + such that { J p : p ∈ Q } is a deltasystem of root ˜ J .Notice that X ˜ J ˜ E has cardinality c , so there exists Q ⊆ Q of cardinality c + such thatfor every p, q ∈ Q and ( n, m, g ) ∈ ˜ J = J p ∩ J q , c pn,m,g = c qn,m,g . Similarly, since ([ ω ] ω ) ˜ J has cardinality c , there exists Q ⊆ Q of cardinality c + such that for every p, q ∈ Q and( n, m, g ) ∈ ˜ J = J p ∩ J q , A pn,m,g = A qn,m,g .Given p, q ∈ Q , a common extension is given by the element r whose components aredefined as follows: E r = E p ∪ E q , α r = α q = α p , G rn,m = G pn,m ∪ G qn,m , A rn,m,g = A sn,m,g and c rn,m,g = c sn,m,g if ( n, m, g ) ∈ J s (where s ∈ { p, q } ).To define φ r , notice that X E p \ ˜ E ⊕ X ˜ E ⊕ X E q \ ˜ E . Let π : X E r → X E p \ ˜ E , π : X E r → X ˜ E , π : X E r → X E q \ ˜ E be the projections. Define φ r = φ p ◦ π + φ p ◦ π + φ q ◦ π = φ p ◦ π + φ q ◦ π + φ q ◦ π . (cid:3) Proposition 4.7.
Let g be sequence of one of the types of X and m >
1. If g is of types 1to 10, let n = 1. If g is type 11, let or n the order of g . Then S n,m,g = { p ∈ P : g ∈ G pn,m } isopen and dense in P . Proof.
Let p ∈ P be an arbitrary condition. Fix E countable such that E p ⊆ E and g ( k ) ∈ X E for each k ∈ ω .Fix µ ∈ C n,m \ E . We set E q = E ∪ { µ } . For each ( m ′ , n ′ ) = ( m, n ) with m ′ , n ′ ≥
1, define G qn ′ ,m ′ = G pn ′ ,m ′ and G qn,m = G pn,m ∪ { g } . Set α q = α p .For every n ′ ≥ m ′ > g ′ ∈ G p,n ′ ,m ′ \ { g } , define c qn ′ ,m ′ ,g ′ = c pn ′ ,m ′ ,g ′ and A qn ′ ,m ′ ,g ′ = A pn ′ ,m ′ ,g ′ . It remains to define c qn,m,g and A qn,m,g . Let c qn,m,g = n χ ( µ, if n > χ µ if n = 1.Extend φ p : X E p → T α p to φ : X E → T α p using divisibility. Now, let A ⊆ ω be an infinitesuch that the sequence ( φ ([ g ( k )]) : k ∈ A ) is convergent, as T α p is a compact metric space.Extend φ to a homomorphism φ q : X E q → T α p such that φ q ([ c q,n,m,g ]) = lim( φ ([ g ( k )]) : k ∈ A ). Set A qn,m,g = A . Then q ≤ p and q ∈ S n,m,g . (cid:3) Theorem 4.8.
Assume CH in the ground model V . Then P preserves cardinals, c and doesnot add reals. If G is generic over P , then the G -generic homomorphism Φ is a well definedinjective homomorphism from X into T c . Moreover, the following holds:(1) For every sequence g of one of the types from 1 to 10 in X and m ≥
1, there exists µ ∈ C ,m such that [ χ µ ] is an accumulation point of (Φ([ g ( k )]) : k ∈ ω )(2) For every sequence g of type 11 and order n in X and for every m ≥
1, there exists µ ∈ C n,m such that [ n χ µ, ] is an accumulation point of ( φ ([ g ( k )]) : k ∈ ω ).(3) (Φ([ N χ n )]) n ∈ ω converges to 0 ∈ X , for every natural N ≥ Proof.
By CH, propositions 4.5 and 4.6, P is ω closed and has the ω chain condition, so P preserves cardinals, does not add reals and preserves c . Notice that since being a type is absolute for transitive models of ZFC, the functions of type 1 to 11 are the same in theground model and in the extension.Let G be a P -generic filter and Φ the associated generic homomorphism.Φ is well defined: suppose p, q ∈ G , ξ < α p ∩ α q and e ∈ X E p ∩ X E q . We must see that φ p ( e )( ξ ) = φ q ( e )( ξ ). Since G is a filter, there exists r such that r ≤ p, q , so ξ < α r , e ∈ X E r and φ p ( e )( ξ ) = φ r ( e )( ξ ) = φ q ( e )( ξ )Now we verify that the domain of Φ is X , that the codomain is T c and that Φ is injectiveat the same time. It is clearly that the domain contained in X . Let e = 0 be an elementof X and α < c . By propositions 4.3 and 4.4, C e and A α are open and dense subsets of P ,therefore there exists p ∈ G such that α p > α e ∈ X E p , φ p ( e ) = 0. So there exists ξ < α p such that φ p ( e )( ξ ) = 0, which implies that Φ( e )( ξ ) = 0. Moreover, α ⊆ dom Φ( e ) ⊆ c . Since α is arbitrary, dom Φ( e ) = c .Φ is an homomorphism: given e, e ′ ∈ X , by 4.3 there exists p ∈ G such that e, e ′ , e + e ′ ∈ X E p . Since φ p is an homomorphism, it follows that Φ( e + e ′ ) = φ p ( e + e ′ ) = φ p ( e ) + φ p ( e ′ ) =Φ( e ) + Φ( e ′ ).Let g be a type and m >
1. If g is of type 1 to 10, let n = 1. If g is type 11, let n be theorder of g . Then by Proposition 4.7, there exists G ∩ S n,m,g . Fix p in this intersection. Weclaim Φ( c p,n,m,g ) is an accumulation point of Φ([ g ]).We know φ p ( c pn,m,g ) is the limit of the convergent sequence ( φ p ([ g ( k )]) : k ∈ A pn,m,g ).Fix F a finite subset of c and let α such that F is a subset of α . Let q ≤ p such that α < α q (which exists since A α +1 ∩ G = ∅ ). Then ( π F ◦ Φ([ g ( k )]) : k ∈ A qn,m,g ) converges to π F ◦ Φ( c qn,m,g ). Since c qn,m,g = c qp,n,m,g , this concludes that Φ( c pn,m,g ) is an accumulation pointof (Φ([ g ( k )]) : k ∈ ω ).It remains to see that (Φ([ n ! S χ n ]) : n ∈ ω ) is a convergent sequence in 0 ∈ T ω . Let ξ < c .Let p ∈ G such that α p > ξ . Then ( π ξ ◦ Φ([ n ! S χ n ]) : n ∈ ω ) = ( π ξ ◦ φ p ([ n ! S χ n ]) : n ∈ ω )converges to 0. Since ξ is arbitrary, we are done. (cid:3) The subspace topology on large subgroups of X.
Of course, not every subgroupof X is countably compact with the forced topology. However, some of them are if they haveenough accumulation points. Thus, we define the concept of large subgroup of X . Definition 4.9.
Let H be a subgroup of X . Let D be the set of all integers n > H contains an isomorphic copy of the group Z ( c ) n .We say that H is a large subgroup of X if 2 c ≥ | G | ≥ r ( G ) ≥ c , for all d, n ∈ N with d | n the group dG [ n ] is either finite or has cardinality at least c and there exist ( k n : n ∈ D ) with k n a positive integer such that:i) { χ µ ∈ X : µ ∈ C } ⊆ H , andii) { [ n χ ( µ, ] ∈ X : n ∈ D, µ ∈ ∪ n ∈ D C n,k n } ⊆ H . (cid:3) Theorem 4.10.
Consider X with the group topology in Theorem 4.8. If H is a largesubgroup of X , then it is countably compact in the subspace topology and has convergentsequences. Proof.
It follows from Theorem 4.8 that if S is a positive integer, then the sequence (cid:0) S n ! χ n : n ∈ ω (cid:1) converges to the neutral element of X . Since both the elements of the ORCING COUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 11 sequence is eventually in H and the limit is in H , it follows that H has non-trivial convergentsequences.Let g : ω → H . Take any ˜ g : ω → X such that [˜ g ] = g. It follows from Proposition 3.1that there exist h : ω → X such that h ∈ H H or [ h ] is a constant in H , c ∈ X with [ c ] ∈ H , F ∈ [ ω ] <ω , p i , q i ∈ Z with q i = 0 for every i ∈ F , ( j i : i ∈ F ) increasing enumerations ofsubsets of ω and j : ω → ω strictly increasing such that˜ g ◦ j = h + c + X i ∈ F p i q i f ◦ j i with q i ≤ j i ( n ) for each n ∈ ω and i ∈ F , where f : ω → X is given by f ( n ) = n ! χ n forevery n ∈ ω .In the case where [ h ] is constant, say constantly v ∈ X , we have that g ◦ j = [˜ g ] ◦ j =[˜ g ◦ j ] = v + [ c ] + P i ∈ F [ p i q i f ◦ j i ]) converges to v + [ c ].In the case h ∈ H H and h is type 11 of order n , then H contain infinitely many copies of Z n . Thus, by hypothesis, n ∈ D . Since n ∈ D , it follows from H H ⊆ H that ( h ( k ) : k ∈ ω )has an accumulation point [ n χ µ, ] with µ ∈ C n,k n . Hence, an accumulation point of h in H .Thus the sequence ( g ◦ j ( k ) : k ∈ ω ) has an accumulation point in [ n χ µ, ] + [ c ] in H .In the case h ∈ H G and h is type 1 to 10, it follows from H G ⊆ H that ( h ( k ) : k ∈ ω ) hasan accumulation point [ χ µ ] with µ ∈ C . Hence, an accumulation point of h in H . Thus thesequence ( g ◦ j ( k ) : k ∈ ω ) has accumulation point [ χ µ ] + [ c ] in H . (cid:3) The classification of Abelian groups of cardinality c . Immersions.
We change slightly the statement and the notation of Proposition 6.1.in [1] to facilitate the application, but it is implicit in the proof in [1].We define A = ( Q / Z ) ( P ) ⊕ Q ( P ) ⊕ U . Definition 5.1.
We say that W is a nice subgroup of W c if there exists a family of positiveintegers ( n ξ ) ξ ∈ P such that W = ( Q / Z ) ( S ξ ∈ P { ξ }× n ξ ) ⊕ Q ( P ) . For this W we denote ~P = ( S ξ ∈ P { ξ } × n ξ ), so W = ( Q / Z ) ( ~P ) ⊕ Q ( P ) . (cid:3) Proposition 5.2.
Let H be an Abelian group such that | H | = r ( H ) = c with H a subgroupof A c .Let D be the set of all integers n > H contains an isomorphic copy of thegroup Z ( c ) n .Then there exist W a nice subgroup of W c , K ∈ [ P ] c with ω ⊆ K , a family ( K n : n ∈ D )of pairwise disjoint elements of [ P ] c , a family ( z ξ : ξ ∈ S n ∈ D K n ) and a group monomorphism φ : A c → W such that:a) { χ ξ ∈ W P : ξ ∈ K } ⊆ φ [ H ],b) { z ξ ∈ W P : ξ ∈ ∪ n ∈ D K n } ⊆ φ [ H ],c) o( z ξ ) = n, ∀ ξ ∈ K n , n ∈ D andd) supp z ξ ⊆ { ξ } × ω ∀ ξ ∈ ∪ n ∈ D K n .We say that φ is a nice immersion for H . (cid:3) This proposition follows from Proposition 6.1. of [1]: W c is naturally isomorphic to thegroup W that appears in [1], and this natural isomorphism preserves the nice subgroups defined in that article. Finally, W is divisible, so we can extend the isomorphism thatProposition 6.1. gives us to the whole group A c . Proposition 5.3.
Let H be an Abelian group such that 2 c ≥ | H | ≥ r ( H ) ≥ h .Let D be the set of all integers n > H contains an isomorphic copy of thegroup Z ( c ) n .Then there exists a family ( k n : n ∈ D ) of positive integers and a group monomorphism ϕ : H → X such that:i) { χ µ ∈ X P ∪ R : µ ∈ C } ⊆ ϕ [ H ], andii) { [ n χ ( µ, ] ∈ X P ∪ R : n ∈ D, µ ∈ ∪ n ∈ D C n,k n } ⊆ ϕ [ H ].Thus, G is isomorphic to a large subgroup of X . Proof.
By theorems 1.8 and 1.9, we may consider H is a subgroup of A . Then we can fix asubgroup ˜ H of H of cardinality c such that r ( ˜ H ) = c and for every n ∈ D , there exists a copyof ( Z n ) c in ˜ H . By a trivial permutation of coordinates we can assume that ˜ H is a subgroup A c . Applying Proposition 5.2, there exist W a nice subgroup of W c , K ∈ [ P ] c with ω ⊆ K ,a family ( K n : n ∈ D ) of pairwise disjoint elements of [ P ] c , a family ( z ξ : ξ ∈ S n ∈ D K n ) anda group monomorphism φ : A c → W c = W P ⊕ W P such that:a) { ( χ ξ ∈ W P : ξ ∈ K } ⊆ φ [ ˜ H ],b) { z ξ ∈ W P : ξ ∈ ∪ n ∈ D K n } ⊆ φ [ ˜ H ],c) o( z ξ ) = n, ∀ ξ ∈ K n , n ∈ D andd) supp z ξ ⊆ { ξ } × ω, ∀ ξ ∈ ∪ n ∈ D K n .We can shrink K n if necessary to find k n positive integer such thate) | supp z ξ | = k n , for each n ∈ D and ξ ∈ K n .By making some permutation within each { ξ } × k n we can further assume thatf) supp z ξ = { ξ } × k n for each n ∈ D and ξ ∈ K n .Define W = W c ⊕ U .We can assume that φ : A c → W and extend it to φ : A = A c ⊕ U → W = W c ⊕ U ,using the identity on U .Fix σ n a bijection between K n and C n,k n for each n ∈ D and σ a bijection between K and C with σ ( k ) = k for every k ∈ ω .Define η : W −→ X an injective homomorphism such that- η : W { ξ }× k n → X { σ n ( ξ ) }× k n is an isomorphism with η ( z ξ ) = [ n χ ( σ n ( ξ ) , ] for each n ∈ D and ξ ∈ K n (this is possible by condition f )),- η ([ χ ξ ]) = [ χ σ ( ξ ) ] for each ξ ∈ K ,- η restricted to U is the identity.Now, let ϕ = η ◦ φ | G . The homomorphism is an embedding, since both η an φ are injectivehomomorphisms.Applying η in a ) it follows that { η ( χ ξ ) ∈ η [ W c ] ⊆ η [ W ] : ξ ∈ K } ⊆ ϕ [ ˜ H ].Therefore, { χ µ ∈ X : µ ∈ C } ⊆ ϕ [ ˜ H ] ⊆ ϕ [ H ] and i ) holds.Likewise, condition ii ) holds. (cid:3) ORCING COUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 13
The classification.Theorem 5.4.
Consider X with the topology from Theorem 4.8. Let H be a group suchthat 2 c ≥ | H | ≥ r ( H ) = c and for all d, n ∈ N with d | n , the group dG [ n ] is either finiteor has cardinality at least c . Then H admits a countably compact group topology with anon-trivial convergent sequence. Proof.
By Proposition 5.3, the group H is isomorphic to a large subgroup of X thereforeby Theorem 4.10, H admits a countably compact group topology with a convergentsequence. (cid:3) We can now (consistently) answer Dikranjan and Shakhmatov’s question for Abeliangroups of cardinality ≤ c . Corollary 5.5.
Consider the forcing model in Theorem 4.8Let H be a non-torsion Abelian group of size at most 2 c . Then the following are equivalent1) ( ⋆ ) 2 c ≥ | H | ≥ r ( H ) = c and for all d, n ∈ N with d | n , the group dH [ n ] is either finiteor has cardinality at least c ;2) H admits a countably compact Hausdorff group topology3) H admits a countably compact Hausdorff group topology with non-trivial convergentsequences. Proof. If H admits a countably compact group topology then condition ( ⋆ ) is satisfied inZFC. Then 2) implies 1).By Theorem 5.4, if H satisfies ( ⋆ ) then 3) holds.Finally 3) implies 2) is obvious. (cid:3) Some comments and questions
So far, the forcings that have been used in constructions of countably compact groupswith some particular properties were the Cohen model, the Random model [20] for groupsof cardinality c and a variation of forcing to construct Kurepa trees [15], [22], [24], [7] and[9] for groups of cardinality at most 2 c .The use of selective ultrafilters for countably compact groups with some property startedfor groups of cardinality c ([26],[13]), then for groups of cardinality 2 c ([16], [23]). To obtainlarger examples, we have used p compactness and the existence of basis in the ultrapower ofa direct sum of torsion groups of bounded order [8] or the direct sum of Q ’s [3].Apparently, selective ultrafilters are not enough to produce a classification of countablycompact groups of cardinality 2 c . Thus, it seems that the best chance to classify the largeAbelian groups that admit a countably compact group topologies is through some other typeof forcing. Problem 6.1.
What other types of forcing can generate countably compact groups withsome property of interest?Further questions in ZFC that are still open after the breakthrough in M. Hrusak, U. A.Ramos-Garcia, J. van Mill and S. Shelah [19] are:
Problem 6.2.
Is there a countably compact group topology on the free Abelian group ofcardinality c in ZFC (with/without non-trivial convergent sequEnces)? Tomita [25] showed in ZFC that if there exists a non-torsion countably compact Abeliangroup without non-trivial convergent sequences then there exists a countably compact freeAbelian group without non-trivial convergent sequences.As it was the case for the classification obtained by Dikranjan and Tkacheko [11], ananswer to the question above is the first step for the following:
Problem 6.3.
Classify in ZFC the Abelian groups of cardinality c that admit a countablycompact group topology.The existence of a countably compact free Abelian group of cardinality c without non-trivial convergent sequences implies the existence of Wallace semigroups (a countablycompact both-sided cancellative semigroup that is not algebraically a group). From theexistence of a countably compact group topology without non-trivial convergent sequencesin ZFC, it is natural to try to answer the following question due to Wallace: Problem 6.4.
Is there a Wallace semigroup in ZFC?The known examples of Wallace semigroups are under CH [17], Martin’s Axiom forcountable posets [21], c incomparable selective ultrafilter [16] and a single selective ultrafilter[5]. Only the one in [21] was not obtained as a semigroup of a countably compact free Abeliangroup without non-trivial convergent sequences. References
1. M. K. Bellini, A. C. Boero, I. Castro-Pereira, V. O. Rodrigues, and A. H. Tomita,
Countably compactgroup topologies on non-torsion abelian groups of size c with non-trivial convergent sequences , TopologyAppl. (2019), Accepted.2. M. K. Bellini, A. C. Boero, V. O. Rodrigues, and A. H. Tomita, Algebraic structure of countably compactnon-torsion Abelian groups of size continuum from selective ultrafilters , preprint, 2019.3. M. K. Bellini, V. O. Rodrigues, and A. H. Tomita, On p -compact group topologies on direct sums of Q ,preprint, 2019.4. A. R. Bernstein, A new kind of compactness for topological spaces , Fund. Math. (1970), 185–193.5. A. C. Boero, I. Castro-Pereira, and A. H. Tomita, Countably compact group topologies on the free abeliangroup of size continuum (and a Wallace semigroup) from a selective ultrafilter , Acta Math. Hungar.(2019), Accepted.6. A. C. Boero, S. Garcia-Ferreira, and A. H. Tomita,
A countably compact free Abelian group of sizecontinuum that admits a non-trivial convergent sequence , Topology Appl. (2012), 1258–1265.7. I. Castro-Pereira and A. H. Tomita,
A countably compact free abelian group whose size has countablecofinality , Appl. Gen. Topol. (2004), 97–101.8. , Abelian torsion groups with a countably compact group topology , Topology Appl. (2010),44–52.9. D. Dikranjan and D. Shakhmatov,
Forcing hereditarily separable compact-like group topologies on Abeliangroups , Topology Appl. (2005), 2–54.10. ,
Selected topics from the structure theory of topological groups , Open problems in topology II(E. Pearl, ed.), Elsevier, 2007, pp. 389–406.11. D. Dikranjan and M. G. Tkachenko,
Algebraic structure of small countably compact Abelian groups ,Forum Math. (2003), 811–837.12. Garcia-Ferreira S. Tomita A. H. Galindo, J., Pseudocompact group topologies with prescribed topologicalsubspaces. , Sci. Math. Jpn. (2009), 269–278.13. S. Garcia-Ferreira, A. H. Tomita, and S. Watson, Countably compact groups from a selective ultrafilter ,Proc. Amer. Math. Soc. (2005), 937–943.14. N. Hindman and D. Strauss,
Algebra in the Stone- ˇCech compactification: Theory and applications , DeGruyter Textbook, De Gruyter, 2011.
ORCING COUNTABLY COMPACT GROUP TOPOLOGIES WITH CONVERGENT SEQUENCES 15
15. P. B. Koszmider, A. H. Tomita, and S. Watson,
Forcing countably compact group topologies on a largerfree Abelian group , Topology Proc. (2000), 563–574.16. R. E. Madariaga-Garcia and A. H. Tomita, Countably compact topological group topologies on free Abeliangroups from selective ultrafilters , Topology Appl. (2007), 1470–1480.17. D. Robbie and S. Svetlichny,
An answer to A D Wallace’s question about countably compact cancellativesemigroups , Proc. Amer. Math. Soc. (1996), 325–330.18. D. J. S. Robinson,
A course in the theory of groups , Springer, 1995.19. M. Hruˇs´ak, J. Van Mill, U. A. Ramos-Gar´ıa,and S. Shelah,
Countably compact groups without non-trivial convergent sequences , preprint avaiablein http://matmor.unam.mx/~michael/preprints_files/Countably_compact.pdf .20. P. J. Szeptycki and A. H. Tomita,
HFD groups in the Solovay model , Topology Appl. (2009),1807–1810.21. A. H. Tomita,
The Wallace problem: a counterexample from MA countable and p -compactness , Canad.Math. Bull. (1996), 486–498.22. , Two countably compact topological groups: one of size ℵ ω and the other of weight ℵ ω withoutnon-trivial convergent sequences , Proc. Amer. Math. Soc. (2003), 2617–2622.23. , A solution to Comfort’s question on the countable compactness of powers of a topological group ,Fund. Math. (2005), 1–24.24. ,
The weight of a countably compact group whose cardinality has countable cofinality , TopologyAppl. (2005), 197–205.25. ,
A van Douwen-like ZFC theorem for small powers of countably compact groups without non-trivial convergent sequences , Topology Appl. (2019), 347–364.26. A. H. Tomita and S. Watson,
Ultraproducts, p -limits and antichains on the Comfort group order ,Topology Appl. (2004), 147–157. Depto de Matem´atica, Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo,Rua do Mat˜ao, 1010 – CEP 05508-090, S˜ao Paulo, SP - Brazil
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